### 1996-29

by Noske, A., Rummler, B..

**Series:** 1996-29, Preprints

- MSC:
- 76F10 Shear flows
- 35Q30 ~Navier-Stokes equations
- 42C15 General harmonic expansions, frames
- 47A75 Eigenvalue problems
- 76H05 Transonic flows

**Abstract:**

We explore the plane parallel channel flows of incompressible Newtonian fluids

in an infinite layer of R 3 with a thickness of h. We presume non-slip conditions

of the fluid at the walls. We write the task as initial-boundary value problem

of the Navier-Stokes equations. Now we transform these equations to a non-

dimensionalized version. The limitation of the domain on an open bounded rect-

angular parallelepiped in R 3 furnished with periodical conditions for the sought

velocity field in the antecedent unbounded directions and the decomposition of

the velocity field in two parts affords equations for the determination of the

remaining velocity. The laminar velocity is fulfilling homogenous Dirichlet con-

ditions. The mainspring of flow is a constant pressure gradient. The application

of Galerkin method precipitates a corresponding system of ordinary differential

equations. The features of this system depend on one parameter Re (Reynolds

number based on the laminar velocity on the midline of the channel). For the

Galerkin method we utilize Stokes eigenfunctions and a fix period 2l = 2 \Theta 2; 69.

For our first considerations we use 356 eigenfunctions (to attain all eigenvalues

less then or equal to 4ß 2 ). For solving our problem in time we exert a Runge-

Kutta-Fehlberg method. We define a kinetic energy as an essential quantity for

the valuation of our solution. The results of computations lead to the conclu-

sions that some features of the laminar turbulent transition are describable by

our Galerkin system.

**Keywords:**

Navier-Stokes equations, plane channel flows, Galerkin method